(1) Show that the following space of infinite sequences c

CLASSICAL APPOXIMATION HOMEWORKS

Due date is November 18, 2019

(1) Show that the following space of infinite sequences c

:= {(x

, x

, . . .) : x

∈ R, lim

x

= 0}

with the norm max

|x

| is not strictly convex.

(2) Find the best approximation in L

([0, 1]) for the function f (x) = x with respect to the space spanned by v

(x) = e

i v

(x) = e

.

(3) Suppose that none of the points a

, a

, . . . , a

is in the interval [a, b]. Show that then

span



1 x − a

, 1

x − a

, . . . , 1 x − a



is a Haar space in C([a, b]).

(4) For what values of a < b the space spanned by the functions (a) {1, cos(x), cos(2x), . . . , cos(nx)}

(b) {sin(x), sin(2x), . . . , sin(nx)}

is a Haar space in C([a, b])?

(5) Let D be the unit sphere,

D = {~ x ∈ R

: k~ xk

= 1 } ⊂ R

.

For what values of s and n one can find Haar spaces of dimension n that are subspaces of C(D)?

(6) Find a polynomial p of degree ¬ 3 that minimizes sup

−1¬x¬1

| |x| − p(x) |.

(7) Find a trigonometric polynomial of the form v(t) = a

+ a

sin t + b

cos t

that best approximates the function sin(t/2) in the uniform norm on the interval [−π, π].

(8) In the set of polynomials p of degree ¬ n such that p(0) = 1 find p

that has minimal uniform norm on [1, 2].

(9) Let p be a polynomial of degee at most n such that kpk

¬ 1. Show that then for any |x| ­ 1 we have |p(x)| ¬ |T

(x)|.

(10) Show that among all the polynomials of degree at most n such that p

(1) = A, the polynomial AT

/n

has the minimal uniform norm in [−1, 1].

Due date is December 9, 2019

(11) Let the operator L : C([a, b]) → C([a, b]) be given as (Lf )(x) =

n

X

i=1

f (x

)g

(x),

1

2

where a ¬ x

< · · · < x

¬ b and g

∈ C([a, b]). Show that L is positive if and only if all the functions g

assume nonnegative values only.

(12) Let L : C([a, b]) → C([a, b]) be a positive linear operator satisfying Lw = w for all polynomials of degree at most 2. Show that then Lf = f for all f ∈ C([a, b]).

(13) Show that if f is a polynomial of degree at most k then the same property possess all the Bernstein polynomials B

f.

(14) Let E be the family of projections L : C([a, b]) → P

, and ˆ E be the family of projections ˆ L : C([−1, 1]) → P

. Show that

L

inf

kL

k = inf

Lˆn∈ ˆE

k ˆ L

k.

(15) Let L : C([−1, 1]) → P

be the interpolation operator corresponding to some points x

, i = 0, 1, 2. Show that

−1¬x0

min

kLk = 5 4 ,

and for the Chebyshev points we have kLk = 5/3. What is kL

k for the equ- ispaced points −1, 0, 1?

(16) Show the following properties of the Chebyshev polynomial T

. (a) (1 − x

)T

(x) − xT

(x) + n

T

(x) = 0

(b) T

(x) = T

(2x

− 1) (c) T

(T

) = T

(17) Let f (x) = x

. Find possibly minimal n such that B

f approximates f with error at most 10

with respect to the uniform norm on [0, 1]?

(18) Show that if f i f

are continuous on [0, 1] then for any  > 0 there is a polynomial p such that kf − pk ¬  and kf

− p

k ¬ , where the norm is uniform on [0, 1].

(19) Let X = L

([0, 1]), f (x) = x

and v

(x) = x

, k = 1, 2, . . . , n, where 0 ¬ m < p

< p

< · · · < p

.

Let V

= span(v

, v

, . . . , v

). Show that dist(f, V

)

= 1

2m + 1

n

Y

k=1

m − p

m + p

+ 1

!2

.

Hint.

det



1 a

+ b

n k,l=1

!

=

Q

k>l

(a

− a

)(b

− b

)

Q

k,l

(a

+ b

) . (20) Let 0 ¬ m < p

< p

< · · · . Show that

n→∞

lim

n

Y

k=1

m − p

m + p

+ 1

!2

= 0 if and only if

∞

X

k=2

1 p

= ∞.

3

(21) (M¨ unz theorem I) Show that the space spanned by the functions v

(x) = x

, where 0 ¬ p

< p

< · · · , is dense in L

([0, 1]) if and only if

1/p

= ∞.

Hint. Use Problems 19 i 20 and the fact that algebraic polynomials are dense in L

([0, 1]).

(22) (M¨ unz theorem II) Show that the space spanned by the functions v

(x) = x

, where 0 ¬ p

< p

< · · · , is dense in C([0, 1]) if and only if p

= 0 and

P∞

k=2

1/p

= ∞. Hint. Use Problem (21).

(23) Is the space spanned by 1, x

, x

, . . . , x

, . . ., where p

are successive primes,

dense in C([0, 1])?